Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 51.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51.a1 | 51a2 | \([0, 1, 1, -59, -196]\) | \(-23100424192/14739\) | \(-14739\) | \([]\) | \(6\) | \(-0.25922\) | |
51.a2 | 51a1 | \([0, 1, 1, 1, -1]\) | \(32768/459\) | \(-459\) | \([3]\) | \(2\) | \(-0.80853\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51.a have rank \(0\).
Complex multiplication
The elliptic curves in class 51.a do not have complex multiplication.Modular form 51.2.a.a
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.